A135828 Expansion of psi(x^2)^8 * (psi(x)^8 + psi(-x)^8) / 2 in powers of x^2 where psi() is a Ramanujan theta function.

1, 36, 378, 2200, 8955, 28836, 78558, 188568, 410805, 828080, 1564686, 2804976, 4809370, 7927380, 12643560, 19594632, 29568204, 43626708, 63094550, 89501040, 124916931, 171803652, 232822908, 311683680, 412601490, 539849556, 699657642, 898801400, 1143680535

Offset: 0

Author

Michael Somos, Nov 29 2007

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 36*x + 378*x^2 + 2200*x^3 + 8955*x^4 + 28836*x^6 + 78558*x^7 + ...
G.f. = q^3 + 36*q^5 + 378*q^7 + 2200*q^9 + 8955*q^11 + 28836*q^13 + 78558*q^15 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A007331, A008774, A045823.

Programs

Magma

Basis( ModularForms( Gamma1(4), 8), 60)[4]; /* Michael Somos, Oct 15 2015 */

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x]^8 (EllipticTheta[ 2, 0, x^(1/2)]^8 + EllipticTheta[ 2, Pi/4, x^(1/2)]^8 16) / 131072, {x, 0, 2 n + 3}]; (* Michael Somos, Oct 15 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, n *= 2; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^24 + eta(x + A)^16 * eta(x^4 + A)^8 ) / ( 2 * eta(x + A)^8 * eta(x^2 + A)^16 / eta(x^4 + A)^16 ), n))};

Formula

Expansion of q^(-3) * ( eta(q^2)^24 + eta(q)^16 * eta(q^4)^8 ) / ( 2 * eta(q)^8 * eta(q^2)^16 / eta(q^4)^16 ) in powers of q^2.

7680 * a(n) = A008774(2*n + 3).

Convolution of A007331 and A045823.